Macroscopic diffusive fluctuations for generalized hard rods dynamics

We study the fluctuations in equilibrium for a dynamics of rods with random length. This includes the classical hard rod elastic collisions, when rod lengths are constant and equal to a positive value. We prove that in the diffusive space-time scaling, an initial fluctuation of density of particles of velocity $v$, after recentering on its Euler evolution, evolve randomly shifted by a Brownian motion of variance $\mathcal D(v)$.